Morita base change in Hopf-cyclic (co)homology
aa r X i v : . [ m a t h . QA ] D ec MORITA BASE CHANGE IN HOPF-CYCLIC (CO)HOMOLOGY
LAIACHI EL KAOUTIT AND NIELS KOWALZIGA
BSTRACT . In this paper, we establish the invariance of cyclic (co)homology of left Hopf algebroids underthe change of Morita equivalent base algebras. The classical result on Morita invariance for cyclic homologyof associative algebras appears as a special example of this theory. In our main application we consider theMorita equivalence between the algebra of complex-valued smooth functions on the classical -torus and thecoordinate algebra of the noncommutative -torus with rational parameter. We then construct a Morita basechange left Hopf algebroid over this noncommutative -torus and show that its cyclic (co)homology can becomputed by means of the homology of the Lie algebroid of vector fields on the classical -torus.
1. I
NTRODUCTION
The concept of left Hopf algebroids provides a natural framework for unifying and extending clas-sical constructions in homological algebra. Group, groupoid, Lie algebra, Lie algebroid and Poisson(co)homology, Hochschild and cyclic homology for associative algebras, as well as Hopf-cyclic homologyfor Hopf algebras, are all special cases of the cyclic homology of left Hopf algebroids since the rings overwhich these theories can be expressed as derived functors are all left Hopf algebroids (see, for example,[4, 8, 9, 22, 23, 24] for more details).As for every (co)homology theory it is an interesting issue to examine its behaviour under (any suitablenotion of) Morita equivalence. Nevertheless, a satisfactory notion of Morita equivalence between two pos-sibly noncommutative left Hopf algebroids is up to our knowledge far from being obvious. The difficultycomes out when, for instance, one tries to understand how the notion of Morita equivalence between twoLie algebroids, in the sense of [10, 15] and others, can be reflected to their respective associated (universal)left Hopf algebroids in such a way that invariant properties, especially homological ones, between equiv-alent Lie algebroids remain invariant at the level of left Hopf algebroids. In the commutative case, that is,for commutative Hopf algebroids, several notions already exist in the literature, see, e.g., [17, 18].In this paper, we restrict ourselves to the case of Morita base change left Hopf algebroids. That is, westudy from a cyclic (co)homology point of view two Morita equivalent left Hopf algebroids of the form ( R, U ) ∼ ( S, ˜ U ) , where R ∼ S are Morita equivalent base rings and ˜ U is constructed from U . It is worthnoticing that for the case of commutative Hopf algebroids or Hopf algebras, this notion reduces to simplychanging the base ring by an isomorphism. Nevertheless, this restriction is not far from geometric appli-cations since, for example, the algebra of smooth functions on a smooth manifold M is Morita equivalentto the endomorphism algebra of global smooth sections of a vector bundle on M . More precisely, one canstart with a smooth vector bundle P → M and a Lie algebroid ( M , E ) , then associate to them a Moritabase change ( C ∞ ( M ) , V Γ( E )) ∼ (End(Γ( P )) , ^ V Γ( E ))) , where ( C ∞ ( M ) , V Γ( E )) is the associated (uni-versal) left Hopf algebroid attached to ( M , E ) , see Section 5. In the aim of illustrating our methods, wegive an explicit application concerning the noncommutative -torus with rational parameter.A left Hopf algebroid ( × R -Hopf algebra) U is, roughly speaking, a Hopf algebra whose ground ring isnot a commutative ring k but a possibly noncommutative k -algebra R , see [3, 32, 36]. In categorical terms, U is a ring extension of the enveloping ring R e = R ⊗ k R o of the base algebra R , where the category ofleft U -modules is a right closed monoidal category, and the forgetful functor to the category of left R e -modules is strict monoidal and preserves right inner hom-functors. As k -bialgebras are underlying Hopf Mathematics Subject Classification.
Primary 16D90, 16E40, 16T05; Secondary 18D10, 16T15, 19D55, 58B34.
Key words and phrases.
Morita equivalence; cyclic homology; Hopf algebroids; vector bundles; Lie algebroids; noncommutativetori.Research of L. El Kaoutit was supported by the grant MTM2010-20940-C02-01 from the Ministerio de Ciencia e Innovaci´on andfrom FEDER. N. Kowalzig acknowledges funding by the Excellence Network of the University of Granada (GENIL). algebras, (left) R -bialgebroids are the underlying structure of (left) Hopf algebroids, but for bialgebroidsthe forgetful functor is in general not right inner-hom preserving.Morita base change for bialgebroids (following [34]) provides a possibility to produce new bialgebroidsby replacing the base algebra R by a Morita equivalent base algebra S in such a way that the resulting R -bialgebroid has a monoidal category of representations equivalent to that of the original R -bialgebroid.More generally, the base algebra R can be replaced by a √ Morita equivalent algebra S , see [37]: twoalgebras are √ Morita equivalent if one has an equivalence of k -linear monoidal categories of bimodules R e Mod ≃ S e Mod . Such an equivalence relation between two bialgebroids is weaker than to consider twobialgebroids to be equivalent if their monoidal categories of (co)representations are so. In particular, Moritabase change establishes a relation between two bialgebroids in a way that is meaningless for ordinary k -bialgebras, as already said before.Apart from what we mentioned above, the importance of the notion of Morita base change moreoverconsists in unifying seemingly different concepts: for example, every weak C -bialgebra (which can be con-sidered as bialgebroids [3, § § R -bialgebroid is left Hopf if and only if its Morita base change equivalent S -bialgebroid is left Hopf aswell [34, Prop. 4.6].In this paper, we will consider the cyclic (co)homology for left Hopf algebroids from [23] and confrontit with the Morita base change theory from [34]. Our aim is to give, in the spirit of [28], the explicit chainmorphisms and chain homotopies that establish equivalences of (co)cyclic modules between the originalleft Hopf algebroid and the Morita base change left Hopf algebroid ˜ U ; see, however, Remark 1 for acomment on a categorical approach. As a consequence, we obtain our central theorem which we copyhere, see the main text for the details and in particular the notation used: Theorem A. (Morita base change invariance of (Hopf-)cyclic (co)homology)
Let ( R, U ) be a left Hopf al-gebroid, M a left U -comodule right U -module which is stable anti Yetter-Drinfel’d, and ( R, S, P, Q, φ, ψ ) a Morita context. Consider its induced √ Morita context ( R e , S e , P e , Q e , φ e , ψ e ) and the Morita basechange left Hopf algebroid ( S, ˜ U := P e ⊗ R e U ⊗ R e Q e ) . Then H • ( U, M ) ≃ H • ( ˜ U , P ⊗ R M ⊗ R Q ) , H • ( U, M ) ≃ H • ( ˜ U , P ⊗ R M ⊗ R Q ) ,HC • ( U, M ) ≃ HC • ( ˜ U , P ⊗ R M ⊗ R Q ) , HC • ( U, M ) ≃ HC • ( ˜ U , P ⊗ R M ⊗ R Q ) are isomorphisms of k -modules. As an application, we first indicate how the classical result of Morita invariance for cyclic homology ofassociative algebras (see, e.g., [7, 12, 28]) fits into our general theory.Second, we consider a Morita context between the complex-valued smooth functions on the commuta-tive real -torus T and the coordinate ring of the noncommutative -torus with rational parameter. Afterreviewing the construction for this case, we apply the Morita invariance to the universal left Hopf algebroidassociated to the Lie algebroid of vector fields on T and its Morita base change left Hopf algebroid g V K over this noncommutative -torus, which establishes a passage from commutative to noncommutative ge-ometry: in the spirit of considering left Hopf algebroids as the noncommutative analogue of Lie algebroidsand their primitive elements as the noncommutative analogue of (generalised) vector fields, the primitiveelements of g V K can be seen to consist of vector fields on the noncommutative torus. Corollary B.
Let q ∈ S be a root of unity, and consider the Lie algebroid (cid:0) R = C ∞ ( T ) , K =Der C ( C ∞ ( T )) (cid:1) of vector fields on the complex torus T and its associated left Hopf algebroid ( R, V K ) .Let M be a right V K -module and ( R, S, P, Q, φ, ψ ) the Morita context of Eq. (5.10) . We then have thefollowing natural C -module isomorphisms H • ( V K, M ) ≃ H • ( g V K, ˜ M ) , HC • ( V K, M ) ≃ HC • ( g V K, ˜ M ) ,H • ( V K, M ) ≃ H • ( g V K, ˜ M ) , HC • ( V K, M ) ≃ HC • ( g V K, ˜ M ) , where g V K is the Morita base change left Hopf algebroid over the noncommutative torus C ∞ ( T q ) whosestructure maps are given as in § ORITA BASE CHANGE IN HOPF-CYCLIC (CO)HOMOLOGY 3
Furthermore, assume that M be R -flat. Then we have that H • ( g V K, ˜ M ) ≃ H • ( K, M ) , HC • ( g V K, ˜ M ) ≃ L i ≥ H • − i ( K, M ) ,H • ( g V K, ˜ M ) ≃ M ⊗ R V • R K, HP • ( g V K, ˜ M ) ≃ L i ≡ • mod2 H i ( K, M ) are natural C -module isomorphisms, where H • ( K, M ) := Tor V K • ( M, R ) , and where HP • denotes peri-odic cyclic cohomology. Acknowledgements.
It is a pleasure to thank J. G´omez-Torrecillas, M. Khalkhali, U. Kr¨ahmer, andA. Weinstein for stimulating discussions and comments. The authors are also grateful to the referee for thecareful reading of the manuscript and the useful comments.2. P
RELIMINARIES
Some conventions.
Throughout this note, “ring” means associative algebra over a fixed commutativeground ring k . All other algebras, modules etc., will have an underlying structure of a central k -module.Given a ring R , we denote by R Mod the category of left R -modules, by R o the opposite ring and by R e := R ⊗ k R o the enveloping algebra of R . An R -ring is a monoid in the monoidal category ( R e Mod , ⊗ R , R ) of R e -modules (i.e., ( R, R ) -bimodules with symmetric action of k ), fulfilling associativity and unitality.Likewise, an R -coring is a comonoid in ( R e Mod , ⊗ R , R ) , fulfilling coassociativity and counitality.Our main object is an R e -ring U . Explicitly, such an R e -ring is given by a k -algebra homomorphism η = η U : R e → U whose restrictions s := η ( − ⊗ k
1) : R → U and t := η (1 ⊗ k − ) : R o → U (2.1)will be called the source and target map, respectively. Left and right multiplication in U give rise to an ( R e , R e ) -bimodule structure on U , that is, four actions of R that we denote by r ✄ u ✁ r ′ := s ( r ) t ( r ′ ) u, r ◮ u ◭ r ′ := us ( r ′ ) t ( r ) , r, r ′ ∈ R, u ∈ U, which are commuting, in the sense that, for every a, a ′ , r, r ′ ∈ R and u, v ∈ U , we have a ′ ◮ (cid:0) r ✄ u ✁ r ′ (cid:1) ◭ a = r ✄ (cid:0) a ′ ◮ u ◭ a (cid:1) ✁ r ′ ; (2.2) (cid:0) u ◭ r (cid:1)(cid:0) v ✁ a (cid:1) = (cid:0) a ◮ u (cid:1)(cid:0) r ✄ v (cid:1) . If not stated otherwise, we view U as an ( R, R ) -bimodule using the actions ✄ , ✁ , denoted ✄ U ✁ . Inparticular, we define the tensor product U ⊗ R U with respect to this bimodule structure. On the other hand,using the actions ◮ , ◭ permits to define the Sweedler-Takeuchi product , see [35, 36]: U × R U := n P i u i ⊗ R v i ∈ U ⊗ R U | P i r ◮ u i ⊗ R v i = P i u i ⊗ R v i ◭ r, ∀ r ∈ R o . One easily verifies that U × R U is an R e -ring via factorwise multiplication, with unit element U ⊗ R U and η U × RU ( r ⊗ k r ′ ) = s ( r ) ⊗ R t ( r ′ ) , for r, r ′ ∈ R .2.2. Bialgebroids. [36] Bialgebroids are a generalisation of bialgebras. An important subtlety is that thealgebra and coalgebra structure are defined in different monoidal categories.
Definition 2.1.
Let R be a k -algebra. A left bialgebroid over R is an R e -ring U together with two homo-morphisms of R e -rings ∆ : U → U × R U, ˆ ε : U → End k ( R ) which turn U into an R -coring with coproduct ∆ (viewed as a map U → U ✁ ⊗ R ✄ U ) and counit ε : U → R , u (ˆ ε ( u ))(1) .So one has for example for u ∈ U , r, r ′ ∈ R ∆( r ✄ u ✁ r ′ ) = r ✄ u (1) ⊗ R u (2) ✁ r ′ , ∆( r ◮ u ◭ r ′ ) = u (1) ◭ r ′ ⊗ R r ◮ u (2) , (2.3)using Sweedler’s shorthand notation u (1) ⊗ R u (2) for ∆( u ) , as well as in U × R U the identity r ◮ u (1) ⊗ R u (2) = u (1) ⊗ R u (2) ◭ r. (2.4)The counit, on the other hand, fulfills for any u, v ∈ U and r, r ′ ∈ Rε ( r ✄ u ✁ r ′ ) = rε ( u ) r ′ , ε ( u ◭ r ) = ε ( r ◮ u ) , ε ( uv ) = ε ( u ◭ ε ( v )) = ε ( ε ( v ) ◮ u ) . (2.5) LAIACHI EL KAOUTIT AND NIELS KOWALZIG
Left Hopf algebroids. [32] Left Hopf algebroids have been introduced by Schauenburg under thename × R -Hopf algebras and generalise Hopf algebras towards left bialgebroids. For a left bialgebroid U over R , one defines the (Hopf-)Galois map β : ◮ U ⊗ Ro U ✁ → U ✁ ⊗ R ✄ U, u ⊗ Ro v u (1) ⊗ R u (2) v, where ◮ U ⊗ Ro U ✁ = U ⊗ k U/ span { r ◮ u ⊗ k v − u ⊗ k v ✁ r | u, v ∈ U, r ∈ R } . (2.6) Definition 2.2. [32] A left R -bialgebroid U is called a left Hopf algebroid (or × R -Hopf algebra ) if β is abijection.By means of a Sweedler-type notation u + ⊗ Ro u − := β − ( u ⊗ R for the translation map β − ( − ⊗ R
1) : U → ◮ U ⊗ Ro U ✁ , one obtains for all u, v ∈ U , r, r ′ ∈ R thefollowing useful identities [32, Prop. 3.7]: u +(1) ⊗ R u +(2) u − = u ⊗ R ∈ U ✁ ⊗ R ✄ U, (2.7) u (1)+ ⊗ Ro u (1) − u (2) = u ⊗ Ro ∈ ◮ U ⊗ Ro U ✁ , (2.8) u + ⊗ Ro u − ∈ U × R o U, (2.9) u +(1) ⊗ R u +(2) ⊗ Ro u − = u (1) ⊗ R u (2)+ ⊗ Ro u (2) − , (2.10) u + ⊗ Ro u − (1) ⊗ R u − (2) = u ++ ⊗ Ro u − ⊗ R u + − , (2.11) ( uv ) + ⊗ Ro ( uv ) − = u + v + ⊗ Ro v − u − , (2.12) u + u − = s ( ε ( u )) , (2.13) u + t ( ε ( u − )) = u, (2.14) ( s ( r ) t ( r ′ )) + ⊗ Ro ( s ( r ) t ( r ′ )) − = s ( r ) ⊗ Ro s ( r ′ ) , (2.15)where in (2.9) we mean the Sweedler-Takeuchi product U × R o U := n P i u i ⊗ Ro v i ∈ ◮ U ⊗ Ro U ✁ | P i u i ✁ r ⊗ Ro v i = P i u i ⊗ Ro r ◮ v i , ∀ r ∈ R o , which is an algebra by factorwise multiplication, but with opposite multiplication on the second factor.Note that in (2.11) the tensor product over R o links the first and third tensor component. By (2.7) and(2.9), one can write β − ( u ⊗ R v ) = u + ⊗ Ro u − v. U -modules. Let ( R, U ) be a left bialgebroid. Left and right U -modules are defined as modules overthe ring U , with respective actions denoted by juxtaposition. We denote the respective categories by U Mod and U o Mod ; while U Mod is a monoidal category, U o Mod in general is not [31]. One has a forgetfulfunctor U Mod → R e Mod using which we consider every left U -module N also as an ( R, R ) -bimodulewith actions anb := a ✄ n ✁ b := s ( a ) t ( b ) n, a, b ∈ R, n ∈ N. (2.16)Similarly, every right U -module M is also an ( R, R ) -bimodule via amb := a ◮ m ◭ b := ms ( b ) t ( a ) , a, b ∈ R, m ∈ M, (2.17)and in both cases we usually prefer to express these actions just by juxtaposition if no ambiguity is to beexpected.2.5. U -comodules. Similarly as for coalgebras, one may define comodules over bialgebroids, but theunderlying R -module structures need some extra attention. For the following definition confer e.g. [31, 2,6]. Definition 2.3. A left U -comodule for a left bialgebroid ( R, U ) is a left comodule of the underlying R -coring ( U, ∆ , ε ) , i.e., a left R -module M with action L R : ( r, m ) rm and a left R -module map ∆ M : M → U ✁ ⊗ R M, m m ( − ⊗ R m (0) satisfying the usual coassociativity and counitality axioms. We denote the category of left U -comodulesby U Comod . ORITA BASE CHANGE IN HOPF-CYCLIC (CO)HOMOLOGY 5
On any left U -comodule one can additionally define a right R -action mr := ε (cid:0) m ( − ◭ r (cid:1) m (0) . (2.18)This action originates in fact from the algebra morphism R → U ∗ , r [ u ε ( u ◭ r )] , (2.19)where U ∗ := Hom R ( U ✁ , R R ) is the right convolution ring of the underlying R -coring U , and the canon-ical functor U Comod → Mod U ∗ that endows any left U -comodule X with a right U ∗ -action givenby x σ = X ( x ) σ ( x ( − ) x (0) for every x ∈ X and σ ∈ U ∗ . The above action is then the restriction to scalars associated to the algebramorphism (2.19), and the action (2.18) is the unique one that turns M into a left R e -module in such a waythat the coaction is an R e -module morphism ∆ M : M → U × R M, where U × R M is the Sweedler-Takeuchi product U × R M := n P i u i ⊗ R m i ∈ U ⊗ R M | P i u i t ( a ) ⊗ R m i = P i u i ⊗ R m i a, ∀ a ∈ R o . In other words, M becomes a left × R - U -comodule. Conversely, any left × R - U -comodule gives rise to aleft U -comodule. This correspondence establishes in fact an isomorphism of categories.As a result of the previous discussion, ∆ M satisfies the identities ∆ M ( rmr ′ ) = (cid:0) r ✄ m ( − ◭ r ′ (cid:1) ⊗ R m (0) , (2.20) m ( − ⊗ R m (0) r = (cid:0) r ◮ m ( − (cid:1) ⊗ R m (0) . (2.21)2.6. Cyclic homology for left Hopf algebroids.
Stable anti Yetter-Drinfel’d modules.
The following definition is the left bialgebroid right moduleand left comodule version of the corresponding notion in [4, 16].
Definition 2.4.
Let ( R, U ) be a left Hopf algebroid, and let M simultaneously be a left U -comodule anda right U -module with action denoted by ( m, u ) mu for u ∈ U , m ∈ M . We call M an anti Yetter-Drinfel’d (aYD) module if:( i ) The two R e -module structures on M originating from its nature as U -comodule resp. right U -module coincide: for all r, r ′ ∈ R , m ∈ Mrm = r ◮ m, (2.22) mr ′ = m ◭ r ′ , (2.23)where the right R -module structure on the left hand side is given by (2.18).( ii ) For u ∈ U and m ∈ M one has the following compatibility between action and coaction: ∆ M ( mu ) = u − m ( − u +(1) ⊗ R m (0) u +(2) . (2.24)The aYD module M is said to be stable (SaYD) if, for all m ∈ M , one has m (0) m ( − = m. (2.25) LAIACHI EL KAOUTIT AND NIELS KOWALZIG
Cyclic (co)homology.
We will not recall the formalism of cyclic (co)homology in full detail; see,e.g., [14, 25] for more information. However, recall that para-(co)cyclic k -modules generalise (co)cyclic k -modules by dropping the condition that the (co)cyclic operator implements an action of Z / ( n + 1) Z onthe degree n part. Thus a para-cyclic k -module is a simplicial k -module ( C • , d • , s • ) and a para-cocyclic k -module is a cosimplicial k -module ( C • , δ • , σ • ) , together with k -linear maps t n : C n → C n resp. τ n : C n → C n satisfying, respectively d i ◦ t n = (cid:26) t n − ◦ d i − if ≤ i ≤ n,d n if i = 0 ,s i ◦ t n = (cid:26) t n +1 ◦ s i − if ≤ i ≤ n,t n +1 ◦ s n if i = 0 , τ n ◦ δ i = (cid:26) δ i − ◦ τ n − δ n if ≤ i ≤ n, if i = 0 ,τ n ◦ σ i = (cid:26) σ i − ◦ τ n +1 σ n ◦ τ n +1 if ≤ i ≤ n, if i = 0 . (2.26)Such a para-(co)cyclic module is called (co)cyclic if t n +1 n = id (resp. τ n +1 n = id ). Any cyclic module C • gives rise to a cyclic bicomplex C •• , see, e.g., [14] for details. The only thing we recall here is that thedifferential on the b -columns is given by b = n X i =0 ( − i d i , (2.27)and likewise β := P n +1 i =0 ( − i δ i for a cocyclic module.2.6.3. The para-(co)cyclic module associated to a left Hopf algebroid ([23], cf. also [24]) . Let M besimultaneously a left U -comodule and a right U -module with compatible left R -action as in (2.22). Set C • ( U, M ) := M ⊗ Ro ( ◮ U ✁ ) ⊗ Ro • , and in each degree n define the following structure maps on it: d i ( m ⊗ Ro x ) = m ⊗ Ro u ⊗ Ro · · · ⊗ Ro (cid:0) ε ( u n ) ◮ u n − (cid:1) m ⊗ Ro · · · ⊗ Ro ( u n − i u n − i +1 ) ⊗ Ro · · · ⊗ Ro u n ( mu ) ⊗ Ro u ⊗ Ro · · · ⊗ Ro u n if i = 0 , if ≤ i ≤ n − , if i = n,s i ( m ⊗ Ro x ) = m ⊗ Ro u ⊗ Ro · · · ⊗ Ro u n ⊗ Ro m ⊗ Ro · · · ⊗ Ro u n − i ⊗ Ro ⊗ Ro u n − i +1 ⊗ Ro · · · ⊗ Ro u n m ⊗ Ro ⊗ Ro u ⊗ Ro · · · ⊗ Ro u n if i = 0 , if ≤ i ≤ n − , if i = n,t n ( m ⊗ Ro x ) = ( m (0) u ) ⊗ Ro u ⊗ Ro · · · ⊗ Ro u n + ⊗ Ro ( u n − · · · u − m ( − ) , (2.28)where we abbreviate x := u ⊗ Ro · · · ⊗ Ro u n . As explained in detail in [23], this cyclic module is thegeneralised “cyclic dual” to the following cocyclic module: set C • ( U, M ) := ( ✄ U ✁ ) ⊗ R • ⊗ R M, with structure maps in degree n given by δ i ( z ⊗ R m ) = ⊗ R u ⊗ R · · · ⊗ R u n ⊗ R mu ⊗ R · · · ⊗ R ∆( u i ) ⊗ R · · · ⊗ R u n ⊗ R mu ⊗ R · · · ⊗ R u n ⊗ R m ( − ⊗ R m (0) if i = 0 , if ≤ i ≤ n, if i = n + 1 ,δ j ( m ) = (cid:26) ⊗ R mm ( − ⊗ R m (0) if j = 0 , if j = 1 ,σ i ( z ⊗ R m ) = u ⊗ R · · · ⊗ R ε ( u i +1 ) ⊗ R · · · ⊗ R u n ⊗ R m ≤ i ≤ n − ,τ n ( z ⊗ R m ) = u − (1) u ⊗ R · · · ⊗ R u − ( n − u n ⊗ R u − ( n ) m ( − ⊗ R m (0) u , (2.29)where we abbreviate z := u ⊗ R · · · ⊗ R u n .In [23] it was shown that, under the minimal assumption (2.22), the maps (2.28) (resp. (2.29)) give riseto a para-cyclic (resp. para-cocylic) module, which is cyclic (resp. cocyclic) if M is SaYD, i.e., additionallyfulfills (2.23)–(2.25). ORITA BASE CHANGE IN HOPF-CYCLIC (CO)HOMOLOGY 7
Let us denote by H • ( U, M ) and HC • ( U, M ) the resulting simplicial and cyclic homology groups of C • ( U, M ) , and likewise by H • ( U, M ) and HC • ( U, M ) the resulting simplicial and cyclic cohomologygroups of C • ( U, M ) .3. √ M ORITA THEORY AND M ORITA BASE CHANGE H OPF ALGEBROIDS
In this section, we first recall some general facts about Morita contexts and their induced √ Morita theoryin the sense of Takeuchi [37]. Secondly, we explain how this theory was used by Schauenburg to introduceMorita base change (left) Hopf algebroids in [34]. In order to establish our main result, we explicitly givehere the structure maps of Schauenburg’s Morita base change (left) Hopf algebroids. From now on, theunadorned symbol ⊗ stands for the tensor product over k , the commutative ground ring.3.1. Morita contexts.
Let R and S be two rings and let S P R and R Q S be two bimodules, together withthe following bimodule isomorphisms: φ : P ⊗ R Q ≃ −→ S, φ − (1 S ) = P p ′ j ⊗ R q ′ j ,ψ : Q ⊗ S P ≃ −→ R, ψ − (1 R ) = P q i ⊗ S p i . (3.1)It is known from Morita theory (see, e.g., [1, p. 60]) that, up to natural isomorphisms, φ and ψ can bechosen in such a way that ( φ ⊗ S P ) = ( P ⊗ R ψ ) and ( ψ ⊗ R Q ) = ( Q ⊗ S φ ) . (3.2)Thus ( R, S, P, Q, φ, ψ ) can be considered as a Morita context. In what follows, we will usually make useof the notation p ′ q ′ := φ ( p ′ ⊗ R q ′ ) and qp := ψ ( q ⊗ S p ) , ∀ p, p ′ ∈ P, q, q ′ ∈ Q. We then have X j p ′ j q ′ j = 1 S , X i q i p i = 1 R , as well as a ( bp ) = ( ab ) p in S P R , b ( aq ) = ( ba ) q in R Q S , for all pairs of elements a, p ∈ P and b, q ∈ Q .The above context is canonically extended to a Morita context between the enveloping rings R e and S e .That is, ( R e , S e , P e , Q e , φ e , ψ e ) is a Morita context as well, where the underlying bimodules are definedby P e := P ⊗ Q o ∈ S e Mod R e ,Q e := Q ⊗ P o ∈ R e Mod S e . Here R o P oS o and S o Q oR o are the opposite bimodules, and φ e , ψ e are the obvious maps. As was argued in[37], this is an induced √ Morita equivalence between R and S , in the sense that the last context inducesa monoidal equivalence between the monoidal categories of bimodules R Mod R and S Mod S . Explicitly,such a monoidal equivalence is set up by the following functors R Mod R ≃ R e Mod P e ⊗ R e − / / S e Mod ≃ S Mod S . Q e ⊗ S e − o o One of the monoidal structure maps of the functor Q e ⊗ S e − is explicitly given by the following naturalisomorphism (cid:0) Q e ⊗ S e X (cid:1) ⊗ R (cid:0) Q e ⊗ S e Y (cid:1) ≃ −→ Q e ⊗ S e ( X ⊗ S Y ) , (cid:0) ( q ⊗ p o ) ⊗ S e x (cid:1) ⊗ R (cid:0) ( b ⊗ a o ) ⊗ S e y (cid:1) ( q ⊗ a o ) ⊗ S e (cid:0) x ( pb ) ⊗ S y (cid:1) , P j (cid:0) ( q ⊗ p ′ jo ) ⊗ S x (cid:1) ⊗ R (cid:0) ( q ′ j ⊗ p o ) ⊗ S y (cid:1) ←− [ ( q ⊗ p o ) ⊗ S e ( x ⊗ S y ) . (3.3)An alternative way of defining these functors is via the following natural isomorphisms: Q e ⊗ S e − ≃ Q ⊗ S − ⊗ S P, P e ⊗ R e − ≃ P ⊗ R − ⊗ R Q. Repeating the same process, we end up with two mutually inverse functors (up to natural isomorphisms) R e Mod R e P e ⊗ R e ( − ) ⊗ R e Q e / / S e Mod S e . Q e ⊗ S e ( − ) ⊗ S e P e o o LAIACHI EL KAOUTIT AND NIELS KOWALZIG
Using the Morita context, this equivalence is canonically lifted to the category of monoids. Thus, if wedenote by R e - Rings the category of R e -rings, i.e., algebra extensions of R e , we have a commutativediagram R e - Rings P e ⊗ R e ( − ) ⊗ R e Q e / / O R (cid:15) (cid:15) S e - Rings O S (cid:15) (cid:15) Q e ⊗ S e ( − ) ⊗ S e P e o o R e Mod R e P e ⊗ R e ( − ) ⊗ R e Q e / / S e Mod S e , Q e ⊗ S e ( − ) ⊗ S e P e o o whose vertical arrows are the forgetful functors. For any R e -ring T we then have functors connecting thecategories of left modules: T Mod P e ⊗ R e ( − ) / / F (cid:15) (cid:15) P e ⊗ R e T ⊗ R e Q e Mod F ′ (cid:15) (cid:15) Q e ⊗ S e ( − ) o o R e Mod P e ⊗ R e ( − ) / / S e Mod . Q e ⊗ S e ( − ) o o (3.4)3.2. Morita base change for left bialgebroids.
In [34], Schauenburg used one of these functors to con-struct a functor from the category of left Hopf algebroids over R to the category of left Hopf algebroidsover S , known as Morita base change left Hopf algebroids . In what follows, we will need an explicit de-scription of this Morita base change left Hopf algebroid structure. So, it will be convenient to review thisconstruction in more detail.Let ( R, S, P, Q, φ, ψ ) be a Morita context. As one can realise from diagram (3.4), the following twoassertions are equivalent:(i) the category of T -modules is a monoidal category and the forgetful functor F is strict monoidal;(ii) the category of ( P e ⊗ R e T ⊗ R e Q e ) -modules is a monoidal category and the forgetful functor F ′ is strict monoidal.Therefore, by Schauenburg’s result [31, Theorem 5.1], starting with a left Hopf algebroid ( R, U ) we canconstruct a new one ( S, ˜ U ) as follows. Denote by ˜ U := P e ⊗ R e U ⊗ R e Q e the image of U . Using the natural isomorphism (3.3) and the diagram (3.4) for the underlying R e -ring U ,we can compute the structure maps of the left Hopf algebroid ( S, ˜ U ) :( i ) Source and target.
Source and target are given by ˜ η : S e −→ P e ⊗ R e U ⊗ R e Q e ,s ⊗ ˜ s o P i, j ( sp ′ j ⊗ q ′ io ) ⊗ R e U ⊗ R e ( q ′ j ⊗ (˜ sp ′ i ) o ) . (3.5)( ii ) Ring structure.
The multiplication in ˜ U is given by ˜ µ : ˜ U ⊗ S e ˜ U −→ ˜ U , ˜ u ⊗ S e ˜ v ( a ⊗ b o ) ⊗ R e (cid:0)(cid:0) u ◭ ( c a ) (cid:1)(cid:0) ( b d ) ✄ v (cid:1)(cid:1) ⊗ R e ( c ⊗ d o ) , (3.6)where ˜ u := (cid:0) ( a ⊗ b o ) ⊗ R e u ⊗ R e ( c ⊗ d o ) (cid:1) and ˜ v := (cid:0) ( a ⊗ b o ) ⊗ R e v ⊗ R e ( c ⊗ d o ) (cid:1) . Theidentity element is given by the image ˜ η (1 S e ) : S e X ( p ′ j ⊗ q ′ io ) ⊗ R e U ⊗ R e ( q ′ j ⊗ p ′ io ) . ( iii ) Coring structure.
The comultiplication is given by ˜∆ : ˜ U −→ ˜ U ⊗ S ˜ U , ˜ u P i, j (cid:0) ( a ⊗ q oi ) ⊗ R e u (1) ⊗ R e ( c ⊗ p ′ jo ) (cid:1) ⊗ S (cid:0) ( p i ⊗ b o ) ⊗ R e u (2) ⊗ R e ( q ′ j ⊗ d o ) (cid:1) , (3.7) ORITA BASE CHANGE IN HOPF-CYCLIC (CO)HOMOLOGY 9 where ˜ u := (cid:0) ( a ⊗ b o ) ⊗ R e u ⊗ R e ( c ⊗ d o ) (cid:1) , and the counit is given by ˜ ε : ˜ U −→ S, ˜ u aε ( u ◭ ( cd )) b. (3.8)( iv ) The left Hopf structure.
The explicit expression for the translation map reads ˜ β − : ˜ U −→ ˜ U ⊗ So ˜ U , ˜ u P i, j (cid:0) ( a ⊗ q ′ jo ) ⊗ R e u + ⊗ R e ( c ⊗ p oi ) (cid:1) ⊗ So (cid:0) ( d ⊗ q oi ) ⊗ R e u − ⊗ R e ( b ⊗ p ′ jo ) (cid:1) , (3.9)where again ˜ u := (cid:0) ( a ⊗ b o ) ⊗ R e u ⊗ R e ( c ⊗ d o ) (cid:1) .3.3. ˜ U -modules and ˜ U -comodules. Consider the diagram analogous to (3.4) for right U -modules. Thefunctor of the first column in that diagram is explicitly given on objects as follows. For M ∈ Mod U , theright ˜ U -module ˜ M := P ⊗ R M ⊗ R Q is equipped with the following action: denote ˜ m := p ⊗ R m ⊗ R q ∈ ˜ M and ˜ u := ( a ⊗ b o ) ⊗ R e u ⊗ R e ( c ⊗ d o ) ∈ ˜ U , and define ˜ m ˜ u := d ⊗ R (cid:0) ( bp ) ◮ m ◭ ( qa ) (cid:1) u ⊗ R c. (3.10)As shown in [34], there is also a monoidal equivalence connecting the categories of left comodules. Moreprecisely, if M ∈ U Comod , then ˜ M is a left ˜ U -comodule with coaction ∆ ˜ M ( ˜ m ) := X i, j (cid:0) ( p ⊗ q oi ) ⊗ R e m ( − ⊗ R e ( q ⊗ p ′ jo ) (cid:1) ⊗ S ( p i ⊗ R m (0) ⊗ R q ′ j ) , (3.11)which exactly coincides with the formula given in [34] in the special case where the left module ✄ U isfinitely generated projective. Lemma 3.1.
Let M be a right U -module and left U -comodule. Then M is aYD (resp. SaYD) if and onlyif ˜ M is.Proof. It is sufficient to prove, say, the direct implication as the opposite direction then follows at oncesince both directions in the Morita base change induced equivalence between U -modules and ˜ U -modulesas well as in the induced equivalence between U -comodules and ˜ U -comodules work the same way.So assume that M is aYD. Then, for any s, t ∈ S , we have ˜ m ˜ η ( t ⊗ s o ) = X i, j (cid:16) p ⊗ R m ⊗ R q (cid:17) (cid:16) (( tp ′ j ) ⊗ q ′ jo ) ⊗ R e U ⊗ R e ( q ′ j ⊗ ( sp ′ i ) o ) (cid:17) (3.10) = X i, j ( sp ′ i ⊗ R (cid:0) ( q ′ i p ) ◮ m ◭ ( qtp ′ j ) (cid:1) ⊗ R q ′ j (2.22) , (2.23) = X i, j ( sp ′ i ⊗ R (cid:0) ( q ′ i p ) m ( qtp ′ j ) (cid:1) ⊗ R q ′ j = X i, j ( sp ′ i q ′ i p ) ⊗ R m ⊗ R ( qtp ′ j q ′ j )= ( sp ) ⊗ R m ⊗ R ( qt ) = s ˜ mt, which gives (2.22) and (2.23) for ˜ M . Now, let us show (2.24) for ˜ M , and start with ˜ m ˜ u = d ⊗ R (cid:0) ( bp ) ◮ m ✁ ( qa ) (cid:1) u ⊗ R c as defined in (3.10). Once computed the coaction of the middle term in thelatter tensor product and taking into account (2.24) for M , apply (3.11) to obtain ∆ ˜ M ( ˜ m ˜ u ) = X i, j h ( d ⊗ q oi ) ⊗ R e (cid:0) ( u − ◭ ( bp ))( m ( − ◭ ( qa )) u +(1) (cid:1) ⊗ R e ( c ⊗ p ′ jo ) i ⊗ S (cid:0) p i ⊗ R m (0) u +(2) ⊗ R q ′ j (cid:1) . On the other hand, using (3.7) and (3.9), we get ˜ u − ˜ m ( − ˜ u +(1) ⊗ S ˜ m (0) ˜ u +(2) = X i ,i ,i ; j ,j ,j h ( d ⊗ q oi ) ⊗ R e (cid:16) (( q i p ′ j ) ◮ u − ◭ ( bp ))(( q i p ′ j ) ◮ m ( − ◭ ( qa )) u +(1) (cid:17) ⊗ R e ( c ⊗ p ′ j o ) i ⊗ S h p i ⊗ R (cid:16) ( m (0) ◭ ( q ′ j p i ))(( q ′ j p i ) ◮ u +(2) ) (cid:17) ⊗ R q ′ j i (2.21) , (2.23) = X i ,i ; j ,j h ( d ⊗ q oi ) ⊗ R e (cid:16) (( q i p ′ j ) ◮ u − ◭ ( bp ))( m ( − ◭ ( qa )) u +(1) (cid:17) ⊗ R e ( c ⊗ p ′ j o ) i ⊗ S h p i ⊗ R (cid:16) ( m (0) )(( q ′ j p i ) ◮ u +(2) ) (cid:17) ⊗ R q ′ j i (2.3) , ( . ) = X i ; j h ( d ⊗ q oi ) ⊗ R e (cid:0) ( u − ◭ ( bp ))( m ( − ◭ ( qa )) u +(1) (cid:1) ⊗ R e ( c ⊗ p ′ j o ) i ⊗ S h p i ⊗ R (cid:16) m (0) u +(2) (cid:17) ⊗ R q ′ j i = ∆ ˜ M ( ˜ m ˜ u ) , where in the last equality we used (2.20) along with (2.22)–(2.24). Analogously one checks the stabilitycondition for ˜ M . (cid:3)
4. M
ORITA BASE CHANGE INVARIANCE IN H OPF - CYCLIC ( CO ) HOMOLOGY
This section contains our main results, Theorems 4.5 & 4.7. More precisely, we construct two mor-phisms between the cyclic modules C • ( U, M ) and C • ( ˜ U , ˜ M ) , where ( S, ˜ U ) is a Morita base change of ( R, U ) , and show that they form quasi-isomorphisms by giving an explicit homotopy. This establishesthe Morita base change invariance for cyclic homology. For the Morita base change invariance of cyclic co homology, we follow the same path although we shall not give the proofs since they are similar to thehomology case.Fix a Morita context ( R, S, P, Q, φ, ψ ) and assume we are given a left Hopf algebroid ( R, U ) , withMorita base change left Hopf algebroid ( S, ˜ U ) as constructed in § § i ,...,n stands for the set of indices { i , · · · , i n } .4.1. The homology case.
Consider the cyclic module (cid:0) C • ( U, M ) , d • , s • , t • (cid:1) as in (2.28). Lemma 4.1.
Let M be a right U -module left U -comodule, subject to both (2.22) and (2.23) . Then thecyclic operator ˜ t : C n ( ˜ U , ˜ M ) → C n ( ˜ U , ˜ M ) for the left Hopf algebroid ˜ U with coefficients in ˜ M isexplicitly given by ˜ t : ˜ m ⊗ So ˜ x X i ,...,n (cid:0) p i ⊗ R m (0) u ⊗ R c (cid:1) ⊗ So (cid:0) ( a ⊗ q oi ) ⊗ R e u ⊗ R e ( c ⊗ p oi ) (cid:1) ⊗ So · · · ⊗ So (cid:0) ( a n ⊗ q oi n − ) ⊗ R e u n + ⊗ R e ( c n ⊗ p oi n ) (cid:1) ⊗ So h ( d n ⊗ q oi n ) ⊗ R e h ( u n − ◭ ( b n d n − ))( u n − − ◭ ( b n − d n − )) · · · ( u − ◭ ( b p )) m ( − i ⊗ R e ( q ⊗ a o ) i , using the notation ˜ m := p ⊗ R m ⊗ R q ∈ ˜ M as well as ˜ x := ˜ u ⊗ So · · · ⊗ So ˜ u n , where ˜ u k := ( a k ⊗ b ok ) ⊗ R e u k ⊗ R e ( c k ⊗ d ok ) for ≤ k ≤ n .Proof. Eq. (2.22) is not directly needed in the computation, but rather to make the operator ˜ t well-defined.By definition we know that ˜ t ( ˜ m ⊗ So ˜ u ⊗ So · · · ⊗ So ˜ u n ) := ˜ m (0) ˜ u ⊗ So ˜ u ⊗ So · · · ⊗ So ˜ u n + ⊗ So (cid:0) ˜ u n − ˜ u n − − · · · ˜ u − ˜ m ( − (cid:1) . Using the formula for the translation map ˜ β in (3.9), we have, along with Eqs. (3.10), (3.11), (2.17), andrepeatedly using the multiplication formula (3.6) ˜ t ( ˜ m ⊗ So ˜ u ⊗ So · · · ⊗ So ˜ u n )= X j ,...,ni ,...,n (cid:16) p i ⊗ R (cid:0) ( q ′ j p i ) ◮ m (0) ◭ ( q ′ j a ) (cid:1) u ⊗ R c (cid:17) ⊗ So (cid:0) ( a ⊗ q ′ j o ) ⊗ R e u ⊗ R e ( c ⊗ p oi ) (cid:1) ⊗ So · · · ⊗ So (cid:0) ( a n ⊗ q ′ j n o ) ⊗ R e u n + ⊗ R e ( c n ⊗ p oi n ) (cid:1) ⊗ So h ( d n ⊗ q oi n ) ⊗ R e ORITA BASE CHANGE IN HOPF-CYCLIC (CO)HOMOLOGY 11 h(cid:0) ( q i n − p ′ j n ) ◮ u n − ◭ ( b n d n − ) (cid:1)(cid:0) ( q i n − p ′ j n − ) ◮ u n − − ◭ ( b n − d n − ) (cid:1) · · · (cid:0) ( q i p ′ j ) ◮ u − ◭ ( b p ) (cid:1) m ( − i ⊗ R e ( q ⊗ p ′ j o ) i = X j ,...,ni ,...,n (cid:16) p i ⊗ R (cid:0) ( m (0) ◭ ( q ′ j a ))( u ✁ ( q ′ j p i )) (cid:1) ⊗ R c (cid:17) ⊗ So (cid:0) ( a ⊗ q ′ j o ) ⊗ R e u ⊗ R e ( c ⊗ p oi ) (cid:1) ⊗ So · · · ⊗ So (cid:0) ( a n ⊗ q ′ j n o ) ⊗ R e u n + ⊗ R e ( c n ⊗ p oi n ) (cid:1) ⊗ So h ( d n ⊗ q oi n ) ⊗ R e h(cid:0) ( q i n − p ′ j n ) ◮ u n − ◭ ( b n d n − ) (cid:1)(cid:0) ( q i n − p ′ j n − ) ◮ u n − − ◭ ( b n − d n − ) (cid:1) · · · (cid:0) ( q i p ′ j ) ◮ u − ◭ ( b p ) (cid:1) m ( − i ⊗ R e ( q ⊗ p ′ j o ) i . By Eqs. (2.23) and (2.21), we can eliminate the sum with the index j . Thus we have ˜ t ( ˜ m ⊗ So ˜ u ⊗ So · · · ⊗ So ˜ u n )= X j ,...,ni ,...,n (cid:16) p i ⊗ R (cid:0) m (0) ( u ✁ ( q ′ j p i )) (cid:1) ⊗ R c (cid:17) ⊗ So (cid:0) ( a ⊗ q ′ j o ) ⊗ R e u ⊗ R e ( c ⊗ p oi ) (cid:1) ⊗ So · · · ⊗ So (cid:0) ( a n ⊗ q ′ j n o ) ⊗ R e u n + ⊗ R e ( c n ⊗ p oi n ) (cid:1) ⊗ So h ( d n ⊗ q oi n ) ⊗ R e h(cid:0) ( q i n − p ′ j n ) ◮ u n − ◭ ( b n d n − ) (cid:1)(cid:0) ( q i n − p ′ j n − ) ◮ u n − − ◭ ( b n − d n − ) (cid:1) · · · (cid:0) ( q i p ′ j ) ◮ u − ◭ ( b p ) (cid:1) m ( − i ⊗ R e ( q ⊗ a o ) i . Repeating the same process, but now using repeatedly (2.9), we can eliminate the sums indexed by i , j , · · · , j n , and obtain the stated formula. (cid:3) In order to show invariance of Hopf-cyclic homology, we will first of all construct a quasi-isomorphism between the b -columns, denoted again by C • ( U, M ) resp. C • ( ˜ U , ˜ M ) , of the cyclic bicom-plexes CC •• ( U, M ) and CC •• ( ˜ U , ˜ M ) associated to the respective cyclic modules (cf. § θ n : C n ( U, M ) → C n ( ˜ U , ˜ M ) as follows: for n = 0 , set θ : M −→ ˜ M , m X i p i ⊗ R m ⊗ R q i , and for n ≥ , abbreviating x := u ⊗ Ro · · · ⊗ Ro u n , set θ n : m ⊗ Ro x X i ,...,n − j ,...,n ( p i ⊗ R m ⊗ R q j ) ⊗ So (cid:0) ( p j ⊗ q oi ) ⊗ R e u ⊗ R e ( q j ⊗ p oi ) (cid:1) ⊗ So · · · ⊗ So (cid:0) ( p j n − ⊗ q oi n − ) ⊗ R e u n ⊗ R e ( q j n ⊗ p oj n ) (cid:1) . (4.1)In the opposite direction, introduce the map γ n : C n ( ˜ U , ˜ M ) → C n ( U, M ) , which is, for n = 0 , γ : ˜ M −→ M, (cid:0) ˜ m := p ⊗ R m ⊗ R q (cid:1) X j ( q ′ j p ) m ( qp ′ j ) , and for n ≥ it is given as γ n : ˜ m ⊗ So ˜ x X j ,...,n m ( qp ′ j ) ⊗ Ro (cid:0) ( q ′ j a ) ✄ u ✁ ( b p ) ◭ ( c p ′ j ) (cid:1) ⊗ Ro (cid:0) ( q ′ j a ) ✄ u ✁ ( b d ) ◭ ( c p ′ j ) (cid:1) ⊗ Ro · · · ⊗ Ro (cid:0) ( q ′ j n d n ) ◮ ( q ′ j n − a n ) ✄ u n ✁ ( b n d n − ) ◭ ( c n p ′ j n ) (cid:1) , (4.2)where ˜ u i := ( a i ⊗ b oi ) ⊗ R e u i ⊗ R e ( c i ⊗ d oi ) ∈ ˜ U for ≤ i ≤ n , and ˜ x := ˜ u ⊗ So · · · ⊗ So ˜ u n . Lemma 4.2.
The maps θ • and γ • are morphisms of chain complexes. Proof.
We only check the compatibility of the differential with γ n since the computation for θ n is similarbut less complicated. Decompose bγ n = =:( i ) z }| { d γ n + =:( ii ) z }| {X ≤ k ≤ n − ( − k d k γ n + =:( iii ) z }| { ( − n d n γ n , where b is the differential (2.27) of the underlying simplicial structure of C • ( U, M ) as in (2.28). Whenapplying this map to an element of the form ˜ m ⊗ So ˜ u ⊗ So · · · ⊗ So ˜ u n (using the notation above), eachterm is explicitly given by ( i ) = X j , ...,n ( m ( qp ′ j ) ⊗ R e (cid:16)(cid:0) ( q ′ j a ) ✄ u ✁ ( b p ) (cid:1) ◭ ( c p ′ j ) (cid:17) ⊗ R e · · · ⊗ R e h ε (cid:16) ( q ′ j n d n ) ◮ (cid:0) ( q ′ j n − a n ) ✄ u n ✁ ( b n d n − ) (cid:1) ◭ ( c n p ′ j n ) (cid:17) ◮ (cid:16)(cid:0) ( q ′ j n − a n − ) ✄ u n − ✁ ( b n − d n − ) (cid:1) ◭ ( c n − p ′ j n − ) (cid:17)i (2.5) = X j , ...,n − ( m ( qp ′ j ) ⊗ R e (cid:16)(cid:0) ( q ′ j a ) ✄ u ✁ ( b p ) (cid:1) ◭ ( c p ′ j ) (cid:17) ⊗ R e · · · ⊗ R e h(cid:16) ( q ′ j n − a n ) ε ( u n ◭ ( c n d n ))( b n d n − ) (cid:17) ◮ (cid:16)(cid:0) ( q ′ j n − a n − ) ✄ u n − ✁ ( b n − d n − ) (cid:1) ◭ ( c n − p ′ j n − ) (cid:17)i ;( ii ) = n − X k =1 X j ,...,n − k − ,n − k +1 ,...,n ( m ( qp ′ j ) ⊗ R e (cid:16)(cid:0) ( q ′ j a ) ✄ u ✁ ( b p ) (cid:1) ◭ ( c p ′ j ) (cid:17) ⊗ R e · · · ⊗ R e h(cid:16) ( b n − k − d n − k ) ◮ (cid:0) ( q ′ j n − k − a n − k ) ✄ u n − k ✁ ( b n − k d n − k − ) (cid:1) ◭ ( c n − k a n − k +1 ) (cid:17)(cid:16) u n − k +1 ◭ ( c n − k +1 p ′ j n − k +1 ) (cid:17)i ⊗ R e · · · ⊗ R e (cid:16) ( q ′ j n d n ) ◮ (cid:0) ( q ′ j n − a n ) ✄ u n ✁ ( b n d n − ) (cid:1) ◭ ( c n p ′ j n ) (cid:17) ;( iii ) = X j ,...,n ( − n (cid:16) m (cid:0)(cid:0) ( qa ) ✄ u ✁ ( b p ) (cid:1) ◭ ( c p ′ j ) (cid:1)(cid:17) ⊗ R e · · · ⊗ R e (cid:16) ( q ′ j n d n ) ◮ (cid:0) ( q ′ j n − a n ) ✄ u n ✁ ( b n d n − ) (cid:1) ◭ ( c n p ′ j n ) (cid:17) . On the other hand, we can also write γ n − ˜ b = =: f ( i ) z }| { γ n − ˜ d + =: g ( ii ) z }| {X ≤ k ≤ n − ( − k γ n − ˜ d k + =: g ( iii ) z }| { ( − n γ n − ˜ d n , where ˜ b is analogously the differential of the underlying simplicial structure of C • ( ˜ U , ˜ M ) . Applying γ n − ˜ b to the same element ˜ m ⊗ So ˜ u ⊗ So · · · ⊗ So ˜ u n , we find that the first term is f ( i ) = X j ,...,n − m ( qp ′ j ) ⊗ R e (cid:16)(cid:0) ( q ′ j a ) ✄ u ✁ ( b p ) (cid:1) ◭ ( c p ′ j ) (cid:17) ⊗ R e · · · ⊗ R e (cid:16) ( q ′ j n − ˆ d n − ) ◮ (cid:0) ( q ′ j n − ˆ a n − ) ✄ ˆ u n − ✁ (ˆ b n − d n − ) (cid:1) ◭ (ˆ c n − p ′ j n − ) (cid:17) , where we denoted the elements ˜ ε (˜ u n ) ◮ ˜ u n − =: (ˆ a n − ⊗ ˆ b n − o ) ⊗ R e ˆ u n − ⊗ R e (ˆ c n − ⊗ ˆ d n − o ) . Com-puting explicitly this term, we obtain ˜ ε (˜ u n ) ◮ ˜ u n − = ˜ u n − ˜ η (1 ⊗ ˜ ε (˜ u n ) o ) (3.5) = X i n , j n ( a n − ⊗ b n − o ) ⊗ R e (cid:16) ( q ′ i n d n − ) ◮ u n − ◭ ( c n − p ′ j n ) (cid:17) ⊗ R e ( q ′ j n ⊗ (˜ ε (˜ u n ) p ′ i n ) o ) (3.8) = X i n , j n ( a n − ⊗ b n − o ) ⊗ R e (cid:16) ( q ′ i n d n − ) ◮ u n − ◭ ( c n − p ′ j n ) (cid:17) ⊗ R e ( q ′ j n ⊗ (cid:0) a n ε ( u n ◭ ( c n d n )) b n p ′ i n ) o (cid:1) = X j n ( a n − ⊗ b n − o ) ⊗ R e (cid:16) u n − ◭ ( c n − p ′ j n ) (cid:17) ⊗ R e ( q ′ j n ⊗ (cid:0) a n ε ( u n ◭ ( c n d n )) b n d n − ) o (cid:1) = ( a n − ⊗ b n − o ) ⊗ R e u n − ⊗ R e (cid:16)(cid:0) c n − ( X j n p ′ j n q ′ j n ) (cid:1) ⊗ (cid:0) a n ε ( u n ◭ ( c n d n )) b n d n − ) o (cid:1)(cid:17) = ( a n − ⊗ b n − o ) ⊗ R e u n − ⊗ R e (cid:16) c n − ⊗ (cid:0) a n ε ( u n ◭ ( c n d n )) b n d n − ) o (cid:1)(cid:17) , thence, ˆ a n − = a n − , ˆ b n − = b n − , ˆ u n − = u n − , and ˆ d n − = a n ε ( u n ◭ ( c n d n )) b n d n − . ORITA BASE CHANGE IN HOPF-CYCLIC (CO)HOMOLOGY 13
Inserting this into the expression of f ( i ) above, one obtains f ( i ) = ( i ) . The second term can be written asfollows: g ( ii ) = n − X k =1 X j ,...,n − k − ,n − k +1 ,...,n ( m ( qp ′ j ) ⊗ R e (cid:16)(cid:0) ( q ′ j a ) ✄ u ✁ ( b p ) (cid:1) ◭ ( c p ′ j ) (cid:17) ⊗ R e · · ·⊗ R e (cid:16)(cid:0) ( q ′ j n − k − a n − k ) ✄ u n − k ✁ ( b n − k d n − k − ) (cid:1) ◭ ( c n − k p ′ j n − k +1 ) (cid:17) ⊗ R e (cid:16)(cid:0) ( q ′ j n − k +1 a n − k +2 ) ✄ u n − k +2 ✁ ( b n − k +2 d n − k ) (cid:1) ◭ ( c n − k +2 p ′ j n − k +1 ) (cid:17) ⊗ R e · · · ⊗ R e (cid:16) ( q ′ j n d n ) ◮ (cid:0) ( q ′ j n − a n ) ✄ u n ✁ ( b n d n − ) (cid:1) ◭ ( c n p ′ j n ) (cid:17) , where we denoted the elements ˜ u n − k ˜ u n − k +1 := ( a n − k ⊗ b n − ko ) ⊗ R e u n − k ⊗ R e ( c n − k ⊗ d n − ko ) (3.6) = ( a n − k ⊗ b on − k ) ⊗ R e (cid:0) ( b n − k +1 d n − k ) ◮ u n − k ◭ ( c n − k a n − k +1 ) (cid:1) u n − k +1 (4.3) ⊗ R e ( c n − k +1 ⊗ d on − k +1 ) . Therefore, g ( ii ) = ( ii ) after substituting (4.3) in g ( ii ) . As for the third term, we have g ( iii ) = X j ,...,n ( − n (cid:16) m (cid:0)(cid:0) ( qa ) ✄ u ✁ ( b p ) (cid:1) ◭ ( c p ′ j ) (cid:1)(cid:17) ⊗ R e · · · ⊗ R e (cid:16) ( q ′ j n d n ) ◮ (cid:0) ( q ′ j n − a n ) ✄ u n ✁ ( b n d n − ) (cid:1) ◭ ( c n p ′ j n ) (cid:17) , which is obviously ( iii ) . We conclude that γ • is a morphism of chain complexes. (cid:3) Proposition 4.3.
The composite γ n θ n is homotopic to the identity, the homotopy h n : C n ( U, M ) → C n +1 ( U, M ) being explicitly given by the following map: for n = 0 , define h : m X i, j m ( q i p ′ j ) ⊗ Ro (( q ′ j p i ) ✄ U ) , and for n ≥ , set h n : m ⊗ Ro x n X k =0 X j ,...,ki ,...,k ( − k + n m ( q i p ′ j ) ⊗ Ro (cid:0) ( q ′ j p i ) ✄ u ◭ ( q i p ′ j ) (cid:1) ⊗ Ro · · · ⊗ Ro (cid:0) ( q ′ j k − p i k − ) ✄ u k ◭ ( q i k p ′ j k ) (cid:1) ⊗ Ro (cid:16) ( q ′ j k p i k ) ✄ U (cid:17) ⊗ Ro u k +1 ⊗ Ro · · · ⊗ Ro u n (4.4) abbreviating x := u ⊗ Ro · · · ⊗ Ro u n as before. Similarly, θ n γ n is homotopic to the identity as well.Proof. We need to check bh = γ θ − id for n = 0 and bh n + h n − b = γ n θ n − id for n > . As for thefirst one, it is immediate that bh ( m ) = X i,j ε (( q ′ j p i ) ✄ U ) ◮ m ( q i p ′ j ) − m = X i,j ( q ′ j p i ) (cid:0) m )( q i p ′ j ) − m = γ θ ( m ) − m. In case n > , since multiplying two consecutive tensor factors of h n kills the respective q, p as well as the q ′ , p ′ between them, it is straightforward to see that n X k =1 ( − k d k h n ( m ⊗ Ro x ) + n − X k =1 ( − k h n − d k ( m ⊗ Ro x ) = 0 . (4.5) As for the remaining terms, we have ( − n +1 d n +1 h n ( m ⊗ Ro x )= − m ⊗ Ro x + ( − n +1 n X k =1 X j ,...,ki ,...,k ( − k + n mu ( q i p ′ j ) ⊗ Ro · · · ⊗ Ro (cid:0) ( q ′ j k − p i k − ) ✄ u k ◭ ( q i k p ′ j k ) (cid:1) ⊗ Ro (cid:16) ( q ′ j k p i k ) ✄ U (cid:17) ⊗ Ro u k +1 ⊗ Ro · · · ⊗ Ro u n = − m ⊗ Ro x + ( − n +1 n − X k =0 X j ,...,ki ,...,k ( − k +( n − ( mu )( q i p ′ j ) ⊗ Ro (cid:0) ( q ′ j p i ) ✄ u ◭ ( q i p ′ j ) (cid:1) ⊗ Ro · · ·⊗ Ro (cid:0) ( q ′ j k − p i k − ) ✄ u k +1 ◭ ( q i k p ′ j k ) (cid:1) ⊗ Ro (cid:16) ( q ′ j k p i k ) ✄ U (cid:17) ⊗ Ro u k +2 ⊗ Ro · · · ⊗ Ro u n = ( − id − ( − n h n − d n )( m ⊗ Ro x ) . Moreover, d h n ( m ⊗ Ro x )= n − X k =0 X j ,...,ki ,...,k ( − k + n ( m ◭ ( q i p ′ j )) ⊗ Ro (cid:0) ( q ′ j p i ) ✄ u ◭ ( q i p ′ j ) (cid:1) ⊗ Ro · · ·⊗ Ro (cid:0) ( q ′ j k − p i k − ) ✄ u k ◭ ( q i k p ′ j k ) (cid:1) ⊗ Ro (cid:16) ( q ′ j k p i k ) ✄ U (cid:17) ⊗ Ro u k +1 ⊗ Ro · · · ⊗ Ro u n − ⊗ Ro (cid:0) ε ( u n ) ◮ u n − (cid:1) + X j ,...,ni ,...,n ( m ◭ ( q i p ′ j )) ⊗ Ro (cid:0) ( q ′ j p i ) ✄ u ◭ ( q i p ′ j ) (cid:1) ⊗ Ro · · · ⊗ Ro (cid:0) ( q ′ j n − p i n − ) ✄ ( q ′ j n p i n ) ◮ u n ◭ ( q i n p ′ j n ) (cid:1) , where we have used Eqs. (2.2) and (2.5). The first sumand is easily seen to be equal to − h n − d and weare left with computing the last sumand: by definition of θ n and γ n (see Eqs. (4.1)–(4.2)) γ n θ n ( m ⊗ Ro x ) = γ n h X k ,...,n − j ,...,n ( p k ⊗ R m ⊗ R q j ) ⊗ So (cid:0) ( p j ⊗ q ok ) ⊗ R e u ⊗ R e ( q j ⊗ p ok ) (cid:1) ⊗ So · · · ⊗ So (cid:0) ( p j n − ⊗ q ok n − ) ⊗ R e u n ⊗ R e ( q j n ⊗ p oj n ) (cid:1)i = X k ,...,n − j ,...,n ; i ,...,n m ( q j p ′ i ) ⊗ R o (cid:0) ( q ′ i p j ) ✄ u ✁ ( q k p k ) ◭ ( q j p ′ i ) (cid:1) ⊗ R o · · · ⊗ R o (cid:0) ( q ′ i n p j n ) ◮ ( q ′ i n − p j n − ) ✄ u n ✁ ( q k n − p k n − ) ◭ ( q j n p ′ i n ) (cid:1) = X j ,...,ni ,...,n ( m ◭ ( q i p ′ j )) ⊗ Ro (cid:0) ( q ′ j p i ) ✄ u ◭ ( q i p ′ j ) (cid:1) ⊗ Ro · · · ⊗ Ro (cid:0) ( q ′ j n − p i n − ) ✄ ( q ′ j n p i n ) ◮ u n ◭ ( q i n p ′ j n ) (cid:1) , where (2.23) was used in the last line and which, as is seen by interchanging the indices, is exactly the lastterm in the expression of d h n ( m ⊗ Ro x ) above. Hence we have shown that d h n ( m ⊗ Ro x ) = (cid:0) − h n − d + γ n θ n (cid:1) ( m ⊗ Ro x ) . Combining this with (4.5), we obtain bh n + h n − b = γ n θ n − id , and this finishes the proof. (cid:3) To pass to the cyclic case, we prove first:
Lemma 4.4.
The morphisms of chain complexes θ • and γ • are morphisms of cyclic objects. That is, theysatisfy: γ • ˜ t • = t • γ • , θ • t • = ˜ t • θ • . ORITA BASE CHANGE IN HOPF-CYCLIC (CO)HOMOLOGY 15
Proof.
We only check the first equation. Take an element ˜ m ⊗ So ˜ u ⊗ So · · · ⊗ So ˜ u n ∈ C n ( ˜ U , ˜ M ) , for n ≥ . Then, applying equations (2.13), (2.15), and (2.20), we can write t n γ n ( ˜ m ⊗ So ˜ u ⊗ So · · · ⊗ So ˜ u n ) = X j ,...,n (cid:16)(cid:0) m (0) ◭ ( q ′ j a ) (cid:1)(cid:0) u ◭ ( c p ′ j ) (cid:1)(cid:17) ⊗ Ro (cid:16) ( q ′ j a ) ✄ u ◭ ( c p ′ j ) (cid:17) ⊗ Ro · · · ⊗ Ro (cid:16) ( q ′ j n − a n ) ✄ u n + ◭ ( c n p ′ j n ) (cid:17) ⊗ Ro h (( q ′ j n d n ) ✄ u n − )(( b n d n − ) ✄ u n − − )(( b n − d n − ) ✄ u n − − ) · · ·· · · (( b d ) ✄ u − )(( b p ) ✄ m ( − ◭ ( qp ′ j )) i . On the other hand, we have γ n ˜ t n ( ˜ m ⊗ So ˜ u ⊗ So · · · ⊗ So ˜ u n ) = X j ,...,n (cid:16)(cid:0) m (0) u (cid:1) ◭ ( c p ′ j ) (cid:17) ⊗ Ro (cid:16) ( q ′ j a ) ✄ u ◭ ( c p ′ j ) (cid:17) ⊗ Ro · · · ⊗ Ro (cid:16) ( q ′ j n − a n ) ✄ u n + ◭ ( c n p ′ j n − ) (cid:17) ⊗ Ro h (( q ′ j n − d n ) ✄ u n − )(( b n d n − ) ✄ u n − − )(( b n − d n − ) ✄ u n − − ) · · ·· · · (( b d ) ✄ u − ) (cid:16) ( b p ) ✄ (cid:0) ( q ′ j n a ) ◮ m ( − ◭ ( qp ′ j n ) (cid:1)(cid:17)i (2.21) = X j ,...,n (cid:16)(cid:0) ( m (0) ◭ ( q ′ j n a )) u (cid:1) ◭ ( c p ′ j ) (cid:17) ⊗ Ro (cid:16) ( q ′ j a ) ✄ u ◭ ( c p ′ j ) (cid:17) ⊗ Ro · · · ⊗ Ro (cid:16) ( q ′ j n − a n ) ✄ u n + ◭ ( c n p ′ j n − ) (cid:17) ⊗ Ro h (( q ′ j n − d n ) ✄ u n − )(( b n d n − ) ✄ u n − − )(( b n − d n − ) ✄ u n − − ) · · ·· · · (( b d ) ✄ u − ) (cid:16) ( b p ) ✄ (cid:0) m ( − ◭ ( qp ′ j n ) (cid:1)(cid:17)i . Now, renumbering the indices we find the equality, and this finishes the proof. (cid:3)
Combining Lemma 4.2, Proposition 4.3, and Lemma 4.4, we conclude that θ • and γ • are in particular equivalences of cyclic modules. Consequently, we can now formulate the main theorem of this paper: Theorem 4.5. (Morita base change invariance of (Hopf-)cyclic homology).
Let ( R, U ) be a left Hopfalgebroid, M a left U -comodule right U -module which is SaYD (i.e., satisfies (2.22) – (2.25) ), and ( R, S, P, Q, φ, ψ ) a Morita context. We then have the following natural k -module isomorphisms: H • ( U, M ) ≃ H • ( ˜ U , P ⊗ R M ⊗ R Q ) ,HC • ( U, M ) ≃ HC • ( ˜ U , P ⊗ R M ⊗ R Q ) . Proof.
This follows at once by using the
SBI sequence for cyclic modules, cf. [25, § (cid:3) The cohomology case.
In this section, we will consider the case of Hopf-cyclic cohomology underMorita base change. Since all steps are basically analogous to the preceding section, we refrain fromspelling out the details and just indicate the main ingredients.Consider the cocyclic module (cid:0) C • ( U, M ) , δ • , σ • , τ • (cid:1) as in (2.29). In the spirit of (4.1) and (4.2), definefirst the map ζ n : C n ( U, M ) → C n ( ˜ U , ˜ M ) as follows: for n = 0 , define ζ : M −→ ˜ M , m X j p ′ j ⊗ R m ⊗ R q ′ j , and for n ≥ , abbreviating y := u ⊗ R · · · ⊗ R u n , define ζ n : y ⊗ R m X i ,...,n − j ,...,n (cid:0) ( p ′ j ⊗ q oi ) ⊗ R e u ⊗ R e ( q ′ j ⊗ p ′ j o ) (cid:1) ⊗ S (cid:0) ( p i ⊗ q oi ) ⊗ R e u ⊗ R e ( q ′ j ⊗ p ′ j o ) (cid:1) ⊗ S · · · ⊗ S (cid:0) ( p i n − ⊗ q oi n − ) ⊗ R e u n ⊗ R e ( q ′ j n − ⊗ p ′ j n o ) (cid:1) ⊗ S ( p i n − ⊗ R m ⊗ R q ′ j n ) . Second, define the map ξ n : C n ( ˜ U , ˜ M ) → C n ( U, M ) , which is ξ : ˜ M −→ M, (cid:0) ˜ m := p ⊗ R m ⊗ R q (cid:1) X i ( q i p ) m ( qp i ) , in degree n = 0 , and for n ≥ is given by ξ n : ˜ y ⊗ S ˜ m X i ,...,n (cid:0) ( q i a ) ✄ ( q i d ) ◮ u ✁ ( b a ) ◭ ( c p i ) (cid:1) ⊗ R (cid:0) ( q i d ) ◮ u ✁ ( b a ) ◭ ( c p i ) (cid:1) ⊗ R · · ·⊗ R (cid:0) ( q i n − d n − ) ◮ u n − ✁ ( b n − a n ) ◭ ( c n − p i n − ) (cid:1) ⊗ R (cid:0) ( q i n d n ) ◮ u n ✁ ( b n p ) ◭ ( c n p i n − ) (cid:1) ⊗ R m ( qp i n ) , where ˜ u i := ( a i ⊗ b oi ) ⊗ R e u i ⊗ R e ( c i ⊗ d oi ) ∈ ˜ U for ≤ i ≤ n , and ˜ y := ˜ u ⊗ S · · · ⊗ S ˜ u n .Third, introduce the homotopy h n : C n +1 ( U, M ) → C n ( U, M ) as follows: in degree n = 0 , set h : u ⊗ R m X i, j ε (cid:0) ( q i p ′ j ) ◮ u (cid:1) m ( q ′ j p i ) , and for n ≥ define h n : y ′ ⊗ R m n X k =0 X j ,...,ki ,...,k ( − k + n u ⊗ R · · · ⊗ R u n − k − ⊗ R ε (cid:0) ( q i p ′ j ) ◮ u n − k (cid:1) ⊗ R (cid:0) ( q i p ′ j ) ◮ u n − k +1 ◭ ( q ′ j p i ) (cid:1) ⊗ R · · · ⊗ R (cid:0) ( q i k p ′ j k ) ◮ u n ◭ ( q ′ j k − p i k − ) (cid:1) ⊗ R m ( q ′ j k p i k ) , (4.6) abbreviating here y ′ := u ⊗ R · · · ⊗ R u n .Now, with the construction of ˜ U and ˜ M as in § C • ( ˜ U , ˜ M ) ; we leave the tedious details to the reader. Similarly as in Lemma 4.2,Proposition 4.3, and Lemma 4.4, one then proves: Lemma 4.6.
The maps ζ • and ξ • are morphisms of cochain complexes, and ξ • ζ • is homotopic to theidentity by means of the homotopy ( ) ; likewise, ζ • ξ • is homotopic to the identity as well. In particular, ζ • and ξ • are equivalences of cocyclic modules. This enables us to conclude:
Theorem 4.7. (Morita base change invariance of (Hopf-)cyclic cohomology).
Let ( R, U ) be a leftHopf algebroid, M a left U -comodule right U -module which is SaYD (i.e., satisfies (2.22) – (2.25) ), and ( R, S, P, Q, φ, ψ ) a Morita context. Then H • ( U, M ) ≃ H • ( ˜ U , P ⊗ R M ⊗ R Q ) ,HC • ( U, M ) ≃ HC • ( ˜ U , P ⊗ R M ⊗ R Q ) are isomorphisms of k -modules.Remark . The proofs of both Theorems 4.7 & 4.5 are based on an explicit construction of (co)chainhomotopies. One could wonder if a more categorical way implicitly leads to the same result but withless computational effort. Closest to our setting is perhaps [4], where a categorical approach to the cyclic(co)homology of bialgebroids was developed based on the notion of admissible septuples. Our situationfits in the particular examples of admissible septuples of [4, Propositions 1.15 & 1.25]; nevertheless, noneof the (co)cyclic objects that, after some additional steps, can be deduced from loc. cit. coincides with our(co)cyclic modules from (2.28) and (2.29), respectively. Since also the involved tensor products differ, thetwo approaches are not even related by considering cyclic duals. That is, our cyclic (co)homology seemsto be different from the one considered in [4].Let us explain how far one can go in applying the approach of [4] to Morita base change invariance.Following the notation of [4], one can show that given a category C , two equivalent categories M , N ,and an admissible septuple S = ( M , C , T l , T r , Π , t , i ) over M , there is an admissible septuple ˜ S =( N , C , ˜ T l , ˜ T r , ˜ Π , ˜ t , ˜ i ) over N , whose corresponding categories of transposition morphisms W S and W ˜ S are also equivalent. Under some natural assumption on C , we know from [4, Corollary 1.11] that thereis a functor ˆ Z ∗ ( S , − ) : W S → ∆ C C to the category of cocyclic objects of C . In this way, the Moritainvariance theory in this context can be interpreted as follows. Consider a transposition morphism ( X, ω ) ∈W S and its image ( ˜ X, ˜ ω ) ∈ W ˜ S . One can then assign to them two cocyclic objects ˆ Z ∗ ( S , ( X, ω )) and ˆ Z ∗ ( ˜ S , ( ˜ X, ˜ ω )) in C . Morita invariance now claims that the associated (co)chain complexes were quasi-isomorphic. At this level of generality, there is seemingly no way which directly furnishes such a quasi-isomorphism if not, analogously to our approach, by manually constructing such a map in special cases(e.g., the aforementioned particular admissible septuples of [4, Propositions 1.15 & 1.25]). ORITA BASE CHANGE IN HOPF-CYCLIC (CO)HOMOLOGY 17
5. A
PPLICATIONS AND E XAMPLES .We give two applications. The first one deals with the well-known Morita invariance of the usualHochschild and cyclic homology for associative algebras. We show that this invariance theory is a con-sequence of our main Theorem 4.5 by applying it to the left Hopf algebroids R e and S e . In the secondapplication we specialise our general results to the Morita context between the complex-valued smoothfunctions on the commutative real -torus T := S × S and the coordinate ring of the noncommuta-tive -torus with rational parameter, establishing thereby a passage from commutative to noncommutativegeometry. We will first review the construction of this context, and next apply the Morita base changeinvariance of the cyclic homology between the left Hopf algebroid attached to the Lie algebroid of vectorfields on T , and the associated Morita base change left Hopf algebroid over this noncommutative -toruswhose structure maps are deduced from § Morita invariance of cyclic homology for associative algebras.
Recall from [32] the left Hopfalgebroid structure of the enveloping algebra R e . Its structure maps are given as follows: s ( r ) := r ⊗ , t ( r o ) := 1 ⊗ r o , ∆( r ⊗ ˜ r o ) := ( r ⊗ ⊗ R (1 ⊗ ˜ r o ) , ε ( r ⊗ ˜ r o ) := r ˜ r , and the inverse of the Hopf-Galoismap is given as ( r ⊗ ˜ r o ) + ⊗ Ro ( r ⊗ ˜ r o ) − := ( r ⊗ ⊗ Ro (˜ r ⊗ . Let now M be a right R e -module which is also an R e -comodule with compatible left R -actions as in(2.22), and denote the coaction by m ( m ′ ( − ⊗ m ′′ ( − ) ⊗ R m (0) , omitting the summation symbol in allwhat follows. Under the isomorphism C • ( R e , M ) = M ⊗ Ro R e ⊗ Ro • ≃ M ⊗ R ⊗ • given by m ⊗ Ro ( r ⊗ ˜ r o ) ⊗ Ro · · · ⊗ Ro ( r n ⊗ ˜ r on ) ˜ r n · · · ˜ r m ⊗ r ⊗ · · · ⊗ r n , (5.1)the para-cyclic operators (2.28) assume the form d i ( m ⊗ y ) = r n m ⊗ r ⊗ · · · ⊗ r n − m ⊗ · · · ⊗ r n − i r n − i +1 ⊗ · · · ⊗ r n mr ⊗ r ⊗ · · · ⊗ r n if i = 0 , if ≤ i ≤ n − , if i = n,s i ( m ⊗ y ) = m ⊗ r ⊗ · · · ⊗ r n ⊗ m ⊗ · · · ⊗ r n − i ⊗ ⊗ r n − i +1 ⊗ · · · ⊗ r n m ⊗ ⊗ r ⊗ · · · ⊗ r n if i = 0 , if ≤ i ≤ n − , if i = n,t n ( m ⊗ y ) = m ′′ ( − m (0) r ⊗ r ⊗ · · · ⊗ r n ⊗ m ′ ( − , (5.2)where we abbreviate y := r ⊗ · · · ⊗ r n , and as before C • ( R e , M ) is cyclic if M is SaYD.Using the isomorphism P e ⊗ R e R e ⊗ R e Q e ≃ −→ S e , ( a ⊗ b o ) ⊗ R e ( r ⊗ ˜ r o ) ⊗ R e ( c ⊗ d o ) φ ( a ⊗ R rc ) ⊗ φ ( d ˜ r ⊗ R b ) o , (5.3)where φ is as in (3.1), together with (3.2) and the isomorphism C • ( S e , ˜ M ) ≃ ˜ M ⊗ S ⊗ n analogously to(5.1), a straightforward computation reveals that the morphism of chain complexes (4.1) reads θ n : m ⊗ y X i ,...,n ( p i ⊗ R m ⊗ R q i ) ⊗ φ ( p i ⊗ R r q i ) ⊗ · · · ⊗ φ ( p i n ⊗ R r n q i ) . In the other direction, we make use of the isomorphism Q e ⊗ S e S e ⊗ S e P e ≃ −→ R e , ( c ⊗ d o ) ⊗ S e ( s ⊗ ˜ s o ) ⊗ S e ( a ⊗ b o ) ψ ( c ˜ s ⊗ S a ) ⊗ ψ ( b ⊗ S sd ) o , together with the inverse of (5.3) given by, cf. (3.5), S e −→ P e ⊗ R e Q e , s ⊗ ˜ s o X i, j ( sp ′ j ⊗ q ′ io ) ⊗ R e ( q ′ j ⊗ (˜ sp ′ i ) o ) , to conclude that the morphism of chain complexes (4.2) becomes here γ n : ( p ⊗ R m ⊗ R q ) ⊗ z X j ,...,n ( ψ ( q ′ j ⊗ S p ) mψ ( q ⊗ S p ′ j ) ⊗ ψ ( q ′ j ⊗ S s p ′ j ) ⊗ · · · ⊗ ψ ( q ′ j n ⊗ S s n p ′ j ) , abbreviating z := s ⊗ · · · ⊗ s n .In a similar manner, one derives the homotopy (4.4) in this case: for n = 0 , we obtain h : m X i,j mψ ( q i ⊗ p ′ j ) ⊗ ψ ( q ′ j ⊗ p i ) , and for n ≥ : h n : m ⊗ y n X k =0 X j ,...,ki ,...,k ( − n + k ( mψ ( q i ⊗ p ′ j )) ⊗ (cid:16) ψ ( q ′ j ⊗ p i ) r ψ ( q i ⊗ p ′ j ) (cid:17) ⊗ · · ·⊗ (cid:16) ψ ( q ′ j n − k − ⊗ p i n − k − ) r k ψ ( q i n − k ⊗ p ′ j n − k ) (cid:17) ⊗ ψ ( q ′ j n − k ⊗ p i n − k ) ⊗ r k +1 ⊗ · · · ⊗ r n , where we abbreviate y := r ⊗ · · · ⊗ r n .One recovers the explicit maps given in [28] for this situation, and hence from Theorem 4.5 the classicalresult [28, 12] of Morita invariance in Hochschild theory follows. In case M = R , with [24, Prop. 3.1], onefurthermore reproduces the classical result of Morita invariance of cyclic homology of associative algebrasfrom [7, 26, 28]: Corollary 5.1.
Let R be an associative k -algebra, M an ( R, R ) -bimodule, and ( R, S, P, Q, φ, ψ ) a Moritacontext. We then have the following natural k -module isomorphism H alg • ( R, M ) ≃ H alg • ( S, P ⊗ R M ⊗ R Q ) , and in case M := R , we obtain HC alg • ( R ) ≃ HC alg • ( S ) . (5.4)Observe that for this corollary no SaYD condition is needed: there is no coaction required to computethe homology of the underlying simplicial object in (5.2) (resp. (2.28)), and for the cyclic homology in (5.4)we only considered the case M := R , with action given by multiplication and coaction R → R e ⊗ R R ≃ R e , r r ⊗ k , which is easily seen to define an SaYD module.5.2. Morita base change invariance in Lie algebroid theory and the noncommutative torus.
Lie algebroids and associated left Hopf algebroids.
Assume that R is a commutative k -algebra (here k is a ground field of characteristic zero) and denote by Der k ( R ) the Lie algebra of all k -linear derivationsof R . Consider a k -Lie algebra L which is also an R -module, and let ω : L → Der k ( R ) be a morphism of k -Lie algebras. Following [30], the pair ( R, L ) is called Lie-Rinehart algebra with anchor map ω , provided ( aX )( b ) = a ( X ( b )) , [ X, aY ] = a [ X, Y ] + X ( a ) Y, for all X, Y ∈ L and a, b ∈ R , where X ( a ) stands for ω ( X )( a ) . A morphism ( R, L ) → ( R, L ′ ) ofLie-Rinehart algebras over R is a map ϕ : L → L ′ of k -Lie algebras such that L ω ●●●●●●●●● ϕ / / L ′ ω ′ { { ✇✇✇✇✇✇✇✇✇ Der k ( R ) is a commutative diagram. These objects form a category which we denote by LieRine ( k , R ) . Example . Here are the basic examples which we will be dealing with, and which motivate the abovegeneral definition:( i ) The pair ( R, Der k ( R )) trivially admits the structure of a Lie-Rinehart algebra.( ii ) A Lie algebroid is a vector bundle
E → M over a smooth manifold, together with a map ω : E → T M of vector bundles and a Lie structure [ − , − ] on the vector space Γ( E ) of global smoothsections of E , such that the induced map Γ( ω ) : Γ( E ) → Γ( T M ) is a Lie algebra homomorphism,and for all X, Y ∈ Γ( E ) and any f ∈ C ∞ ( M ) one has [ X, f Y ] = f [ X, Y ] + Γ( ω )( X )( f ) Y .Then the pair ( C ∞ ( M ) , Γ( E )) is obviously a Lie-Rinehart algebra. ORITA BASE CHANGE IN HOPF-CYCLIC (CO)HOMOLOGY 19
Associated to any Lie-Rinehart algebra ( R, L ) there is a universal object denoted by ( R, V L ) , see [30,19]. Using the notion of smash product (or, more general, of distributive law between two algebras ), wegive here an alternative construction (of which the Massey-Peterson algebra in [27, 19] is a special case) ofthis object: let
U L be the universal enveloping algebra of L with its canonical Hopf algebra structure, andconsider the k -linear map L ω −→ Der k ( R ) . Extending this map to U L , we obtain the structure of a
U L -module algebra on R . Following [35, pp. 117–118], the smash product R U L admits the structure of aleft R -bialgebroid, where the source and the target map coincide. Now take the following factor R -algebraof R U L : π : R U L −→ V L := R U L J L , where J L := h a X − aX i a ∈ R, X ∈ L is the two sided ideal generated by the set { a X − aX } a ∈ R, X ∈ L . The R -bialgebroid structure of R U L projects to V L (see [38]), and by [22, § V L carries a left Hopf algebroidstructure, the translation map on generators a ∈ R , X ∈ L of V L being given by a + ⊗ Ro a − := a ⊗ Ro , X + ⊗ Ro X − := X ⊗ Ro − ⊗ Ro X, where V L ⊗ Ro V L := ◮ V L ⊗ Ro V L ✁ is as in (2.6) (which is why we stick to the symbol R o although R is commutative), and where we identify the elements of R and L with their respective images by theuniversal maps ι R : R → V L , a a J L and ι L : L → V L , X X + J L .5.2.2. Vector bundles versus √ Morita theories.
Let R be a commutative k -algebra as in § P R which is faithful over R . Then it is well known(see, for example, [11, Corollary 1.10]) that R is Morita equivalent to the endomorphism ring End( P R ) since R is commutative. The context maps are given by φ : P ⊗ R P ∗ ≃ −→ End( P R ) , ( p ⊗ σ [ p ′ pσ ( p ′ )]) ,ψ : P ∗ ⊗ End( PR ) P ≃ −→ R, ( σ ⊗ p σ ( p )) , where P ∗ = Hom( P R , R R ) .Following [20, Example 2.3.3], we apply this Morita context to the situation where R is the algebra ofsmooth functions over a manifold M . By the Serre-Swan theorem, it is well known that for a (complex)smooth vector bundle π : P → M of constant rank ≥ the global smooth sections P := Γ( P ) form afinitely generated projective module over the commutative ring R := C ∞ ( M ) of complex-valued smoothfunctions on M , see, for instance, [29, Remark, p. 183]. One can furthermore show [5, Remarque 2,p. 145] that P is of constant rank ≥ (the rank of π ), and as such, P becomes a faithful R -module (asfollows from [5, p. 142, Corollaire, & p. 143, Th´eor`eme 3 (ii)]). Therefore, C ∞ ( M ) is Morita equiva-lent to the endomorphism algebra End( P C ∞ ( M ) ) ≃ Γ(End( P )) . In this way, there is a functor from thecategory LieRine ( C , C ∞ ( M )) to the category of left Hopf algebroids over End( P C ∞ ( M ) ) . This functoris defined on objects by sending any complex Lie-Rinehart algebra ( L, R ) to the left End( P R ) -Hopf al-gebroid P e ⊗ R e V L ⊗ R e Q e , where P e , Q e are defined as in § ( R, End( P R ) , P, P ∗ , φ, ψ ) , i.e., with Q = P ∗ . Remark . An analogue to the previous functor can, in fact, descend to the category of Lie algebroids overa smooth manifold M if we take a real vector bundle and the algebra C ∞ ( M , R ) of real-valued smoothfunctions instead of C ∞ ( M ) = C ∞ ( M , C ) . We know from Example 5.2(ii) that there is a canonicalfaithful functor from the category of Lie algebroids over M to the category of real Lie-Rinehart algebrasover C ∞ ( M , R ) . Now we can compose this functor with the one constructed by the same process asin § LieRine ( R , C ∞ ( M , R )) and LieRine ( C , C ∞ ( M , C )) , except perhaps when M is an almost complex manifold (i.e., a smooth manifoldwith a smooth endomorphism field J : T M → T M satisfying J x = − id T M x for all x ∈ M ).Let us mention that due to our interest in the noncommutative torus, we have been forced to extend thebase field by using the complex-valued functions instead of real-valued ones. The material of the following subsection will appear well known to the reader who is familiar withnoncommutative differential geometry techniques. For the convenience of the rest of the audience, weinclude a detailed exposition following ideas from [13, § § Noncommutative torus revisited.
Consider the Lie group S = { z ∈ C \ { } | | z | = 1 } as a real -dimensional torus by identifying it with the additive quotient R / π Z . Likewise, the real d -dimensionaltorus T d := S × · · · × S is identified with R d / π Z d . The complex algebra of all smooth complex-valuedfunctions on T will be denoted by C ∞ ( T ) .Fix a root of unity q ∈ S and take N ∈ N to be the smallest natural number such that q N = 1 . Let usconsider the semidirect product group G := Z N ⋉ S where Z N = Z /N Z , and operation ( m, n, θ )( m ′ , n ′ , θ ′ ) := ( m + m ′ , n + n ′ , θθ ′ q mn ′ ) , for every pair of elements ( m, n, θ ) , ( m ′ , n ′ , θ ′ ) ∈ G . There is a right action of the group G on the torus T given as follows: ( x , y , z )( m, n, θ ) := ( q m x , q n y , θ zy m ) , ( x , y , z ) ∈ T , ( m, n, θ ) ∈ G . Now, we can show that the map p : T −→ T , ( x , y , z ) ( x N , y N ) satisfies:( i ) p is a surjective submersion;( ii ) G acts freely on T and the orbits of this action coincide with the fibres of p .As a consequence and by applying [21, Lemma 10.3], we see that ( T , p , T , G ) is a principal fibre bundle.We then want to associate a non-trivial vector bundle to the trivial bundle T × C N → T . So, we need toextend the G -action on T to T × C N , which is possible by considering the following left G -action on C N G −→
End C ( C N ) , ( m, n, θ ) (cid:8) ω θU n V − m ω (cid:9) , where U , V are the ( N × N ) -matrices U = · · · · · · ... . . . . . . ... · · · · · · , V = · · · · · · q · · ·
00 0 q · · · ... . . . . . . ... · · · q N − , which satisfy the relations U V = q V U , U N = V N = I N . (5.5)Therefore, we have a right G -action on T × C N defined by (cid:0) ( x , y , z ); ω (cid:1) ( m, n, θ ) := (cid:16) ( x , y , z )( m, n, θ ); ( m, n, θ ) − ω (cid:17) = (cid:16) ( q m x , q n y , θ zy m ); θ − U − n V m ω (cid:17) . The orbit space ( T × C N ) / G = T × G C N with elements u × G ω will be denoted by E q . Notice that bydefinition one has the following formula: ( ug ) × G ω = u × G ( gω ) , for every u ∈ T , ω ∈ C N , and g ∈ G . By applying [21, Theorem 10.7, § T × C N → T , that is, there is a morphism of vector bundles T × C N / / pr (cid:15) (cid:15) E q p (cid:15) (cid:15) ✤✤✤ T p / / T . (5.6)By the results of § C ∞ ( T ) is Morita equivalent to End(Γ( E q )) ≃ Γ (cid:0) End( E q ) (cid:1) .Now, using [21, Theorem 10.12], Γ( E q ) is identified with the G -equivariant subspace C ∞ ( T , C N ) G of C ∞ ( T , C N ) , that is, those f ∈ C ∞ ( T , C N ) for which f ( ug ) = g − f ( u ) , for every u ∈ T , g ∈ G .Hence, we have an isomorphism Γ( E q ) ≃ C ∞ ( T , C N ) G of C ∞ ( T ) -modules. ORITA BASE CHANGE IN HOPF-CYCLIC (CO)HOMOLOGY 21
Next, we want to describe the noncommutative complex algebra
End(Γ( E q )) ≃ Γ (cid:0) End( E q ) (cid:1) . Observethat there is a left Z N -action on the ( N × N ) -matrix algebra M N ( C ) with complex entries, defined by ( m, n ) A := U n V − m AV m U − n , for every A ∈ M N ( C ) , ( m, n ) ∈ Z N . (5.7)There is also a free right Z N -action on T given by ( x , y )( m, n ) := ( q m x , q n y ) , for every ( x , y ) ∈ T , ( m, n ) ∈ Z N . As before, one can construct the orbit space T × Z N M N ( C ) after extending these actions to the trivialalgebra bundle T × M N ( C ) . It turns out that the endomorphism algebra bundle End( E q ) is isomorphic tothis orbit space, and clearly Γ(End( E q )) consists of Z N -equivariant sections, that is, T ∈ Γ(End( E q )) if and only if T ( q m x , q n y ) = ( m, n ) T ( x , y ) , (5.8)for every ( x , y ) ∈ T and ( m, n ) ∈ Z N , where on the right hand side we mean the action (5.7).On the other hand, it is well known that C ∞ ( T ) can be identified with the algebra of all smooth func-tions on R that are π -periodic w.r.t. each of their arguments. By Fourier expansion C ∞ ( T ) consists ofall functions f = X ( k, l ) ∈ Z f k,l u k v l , where { f k,l } ( k, l ) ∈ Z is any rapidly decreasing sequence of complex numbers, that is, for every r ∈ N , theseminorm k f k r = sup k, l ∈ Z (cid:16) | f k,l | (1 + | k | + | l | ) r (cid:17) < ∞ , (5.9)and where u = e i πt , v = e i πs are the coordinate functions on the torus T .It is also well known that the complex matrix algebra M N ( C ) is generated as C -algebra by the elements U , V . Thus, Eqs. (5.5) and (5.8) force any T ∈ Γ(End( E q )) to be of the form T = X k,l, ∈ Z T k, l ( uU ) k ( vV ) l , with coefficients { T k, l } Z satisfying Eq. (5.9). Therefore, there is now a C -algebra isomorphism Γ(End( E q )) → C ∞ ( T q ) , (cid:0) ( uU ) U, ( vV ) V (cid:1) , where C ∞ ( T q ) refers to the complex noncommutative -torus whose elements are formal power Laurentseries in U, V with a rapidly decreasing sequence of coefficients (cf. (5.9)), subject to
U V = q V U . Inconclusion, we have the Morita context ( C ∞ ( T ) , C ∞ ( T q ) , Γ( E q ) , Γ( E q ) ∗ ) , where in addition C ∞ ( T ) and C ∞ ( T q ) are related by the algebra map C ∞ ( T ) −→ C ∞ ( T q ) , ( u U N , v V N ) . In the next subsection, we will use the Morita context stated above together with Theorems 4.5 and4.7 to prove the Morita invariance of both cyclic homology and cohomology from the left Hopf algebroidattached to the Lie algebroid of vector fields over the classical -torus to the associated Morita base changeleft Hopf algebroid over the noncommutative -torus (the primitive elements of which can be seen to consistof noncommutative vector fields, cf. the comment in the Introduction), using the construction performed in § The cyclic homology for the left Hopf algebroid over the noncommutative torus.
Now we willdirect our attention to the Morita invariance of the cyclic homology between the trivial Lie algebroid (cid:0) C ∞ ( T ) , K := Der C (cid:0) C ∞ ( T ) (cid:1)(cid:1) and its induced left Hopf algebroid ( S, g V K := P e ⊗ R e V K ⊗ R e Q e ) ,where R := C ∞ ( T ) , S := C ∞ ( T q ) , P := Γ( E q ) , Q := Γ( E q ) ∗ , (5.10)and where the notation is that of § g V K by using the general description of § P which can beextracted from the dual basis of the trivial bundle T × C N , see Eq. (5.6). Applying Theorems 4.5 & 4.7as well as [23, Theorem 5.2] (and its dual version, cf. [24, Theorem 3.14]), we obtain Corollary 5.3.
Let q ∈ S be a root of unity, and consider the Lie algebroid ( R, K ) of vector fieldsof the complex torus T and its induced left Hopf algebroid ( R, V K ) . Let M be a right V K -moduleand ( R, S, P, Q, φ, ψ ) the Morita context of Eq. (5.10) . We then have the following natural C -moduleisomorphisms H • ( V K, M ) ≃ H • ( g V K, ˜ M ) , HC • ( V K, M ) ≃ HC • ( g V K, ˜ M ) ,H • ( V K, M ) ≃ H • ( g V K, ˜ M ) , HC • ( V K, M ) ≃ HC • ( g V K, ˜ M ) , where g V K is the Morita base change left Hopf algebroid over the noncommutative torus C ∞ ( T q ) .Furthermore, assume that M be R -flat. Then we have that H • ( g V K, ˜ M ) ≃ H • ( K, M ) , HC • ( g V K, ˜ M ) ≃ L i ≥ H • − i ( K, M ) ,H • ( g V K, ˜ M ) ≃ M ⊗ R V • R K, HP • ( g V K, ˜ M ) ≃ L i ≡ • mod2 H i ( K, M ) are natural C -module isomorphisms, where H • ( K, M ) := Tor V K • ( M, R ) , and where HP • denotes peri-odic cyclic cohomology (see, e.g., [25, § for the definition of HP ).Remark . In [39, Theorem 5.2], the Hochschild cohomology of the algebra R = C ∞ ( T ) was computedin terms of the exterior algebra of a two-dimensional complex vector space. So we can apply Corollary5.1 to deduce the Hochschild cohomology of the noncommutative torus C ∞ ( T q ) , where q is not a root ofunity. On the other hand, the same result [39, Theorem 5.2] shows that K = Der C (cid:0) C ∞ ( T ) (cid:1) is a free R -module of rank . One can therefore also consider another application of Theorems 4.5 & 4.7 by taking thecanonical Morita context ( R, M ( R ) , K, K ∗ , db, ev ) and the left Hopf algebroid ( R, V K ) . Here M ( R ) denotes the (4 × -matrices over R , whereas ev : K ∗ ⊗ M R ) K → R, ϕ ⊗ M R ) x ϕ ( x ) stands for theevaluation map and db : K ⊗ R K ∗ → End(K R ) ∼ = M (R) for the dual basis map which sends any element x ⊗ R ϕ to the (4 × -matrix attached to the R -linear map [ y xϕ ( y )] . The details of this application areleft to the reader. R EFERENCES1. Bass, H.:
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